We obtain a complete classification of four-dimensional conformally flat homogeneous pseudo-Riemannian manifolds.
Introduction.Conformally flat manifolds are a classical field of investigation in pseudo-Riemannian geometry. In this framework, it is a natural problem to classify conformally flat homogeneous pseudo-Riemannian manifolds. A conformally flat (locally) homogeneous Riemannian manifold is (locally) symmetric [17]. Hence, it admits as univeral covering either a space form R n , S n (k),In pseudo-Riemannian settings, the problem of classifying conformally flat homogeneous manifolds is more complicated and interesting. Three-dimensional examples were classified independently in [8] and [3], showing the existence of non-symmetric examples. Using the general results introduced in [8], the same authors contributed in [9] to solve the classification problem for Lorentzian manifolds of any dimension, under some assumptions on the structure of the eigenvalues of the Ricci operator of such a manifold. Up to our knowledge, no classification results have been obtained yet for metrics of different signatures, except for the cases with a diagonalizable Ricci operator [8].In the present paper we shall provide a complete classification of four-dimensional conformally flat homogeneous pseudo-Riemannian manifolds. A fundamental step for this classification will be to understand which forms (Segre types) of the Ricci operator, and under which restrictions, may exist for conformally flat pseudo-Riemannian manifolds. As we shall see, nondegenerate forms of the Ricci operator can only occur when a conformally flat homogeneous pseudo-Riemannian four-manifold is (locally) isometric to some Lie group, while the possible degenerate forms are also realized by some homogeneous spaces with nontrivial isotropy, for which we can use Komrakov's classification [10] to deduce all possible examples.Because of the results obtained in [9], we shall focus mainly on the case of a pseudo-Riemannian metric of neutral signature. However, we shall also explain how our results apply to the Lorentzian case.The paper is organized in the following way. In Section 2, we report some basic information on four-dimensional conformally flat pseudo-Riemannian manifolds and describe
We investigate the geometric properties of four-dimensional non-reductive pseudoRiemannian manifolds admitting an invariant metric of signature (2, 2). In particular, we obtain the classification of Walker structures, self-dual and anti-self-dual metrics and para-Hermitian structures for all the invariant metrics of these manifolds. For the examples admitting an invariant parallel null plane distribution, we shall also obtain an explicit description of the invariant metrics in canonical Walker coordinates.
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